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Design of Spatial Experiments: Model Fitting and Prediction

Valerii V. Fedorov

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Computer Science and Mathematics Division

Mathematical Sciences Section

DESIGN OF SPATIAL EXPERIMENTS: MODEL FITTING AND

PREDICTION

Valerii V. Fedorov

Mathematical Sciences Section Computer Science and Mathematics Division Oak Ridge National Laboratory P. 0. Box 2008 Oak Ridge, Tennessee 37831-6367

Date Published: March 1996

Research sponsored by the Laboratory Directed Re- search and Development Program of Oak Ridge National Laboratory

Prepared by the Oak Ridge National Laboratory

Oak Ridge, Tennessee 37831 managed by

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U.S. DEPARTMENT OF ENERGY under Contract No. DE-ACO5-96OR22464

.

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DESIGN OF SPATIAL EXPERIMENTS ... 111

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Standard design problem . . . . . . . . . . . . . . . . . . . . . . . . . . v Optimal designs with bounded density . . . . . . . . . . . . . . . . . . x Correlated observational errors . . . . . . . . . . . . . . . . . . . . . . . xiv Random coefficients regression models: Trend estimation . . . . . . . . xviii Random coefficient regression models: Prediction . . . . . . . . . . . . xxv Comparison with the methods based on the variance - covariance struc- ture of observed random fields . . . . . . . . . . . . . . . . . . . . . . . xxix

xl Unknown covariance function . . . . . . . . . . . . . . . . . . . . . . . xliii Discrete case. Optimality criteria and the lower bounds . . . . . . . . .

10 Space and time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlviii 11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Abstract

The main objective of the paper is to describe and develop model ori-

ented methods and algorithms for the design of spatial experiments. Unlike

many other publications in this area, the approach proposed here is essen-

tially based on the ideas of convex design theory.

1. Introduction

Since the earliest days of the experimental design theory, a number of concepts like

split plots, strips, blocks, Latin squares, etc. (see Fisher (1947)), were strongly

related to experiments with spatially distributed or allocated treatments and ob-

servations. In this survey we confine ourself to what can be considered as an

intersection of ideas developed in the areas of response surface design of experi-

ments and spatial statistics.

The results which we are going to consider are also related to the results de-

veloped by Cambanis (1985), Cambanis and Su (1993), Matern (1986), Micchelli

and Wahba (1981), Sacks and Ylvisaker (1966, 1968, 1970) and Ylvisaker (1975,

1v

1987). What differs in the approach of this paper from those cited? We intend

to use the techniques which are based on the concept of regression models while

the cited studies are based on the ideas developed in the theory of stochastic

processes and the theory of integral approximation.

If this survey were to be written for a very applied audience, the title “optimal

allocations of sensors” or “optimal allocation of observing stations” could be

more appropriate. Environmental monitoring, meteorology, surveillance, some

industrial experiments and seismology are the most typical areas in which the

considered results may be applied. What are the most common features of the

experiments to be discussed?

1. There are variables x E X c Rk, which can be controlled. Usually k = 2 , and in the observing station problem, x1 and 22 are coordinates of stations

and X is a region where those stations may be allocated.

2. There exists a model describing the observed response(s) or dependent vari-

able(s) y. More specifically y and x are linked together by a model, which

may contain some stochastic components.

3. An experimenter or a practitioner can formulate the quantitative objective

function.

4. Once a station or a sensor is allocated a response y can be observed either

continuously or according to any given time schedule without any additional

significant expense.

5. Observations made at different sites may be correlated.

Assumptions 1 - 5 are very loosely formulated and they will be justified when

needed. In the subsequent sections the term ‘‘sensor” stands for what could be

an observing station, meteorological station, radiosonde or well in the particular

applied problem.

DESIGN OF SPATIAL EXPERIMENTS V

2. Standard design problem

In what follows we will mostly refer to experiments which are typical in environ-

mental monitoring setting as a background for the exposition of the main results.

We hope that the reader will be able to apply the ideas and techniques to other

types of experiments.

When assumptions 4 and 5 are not considered we have what will be be

called, the “standard design problem”. The problem was extensively discussed

(see for instance, Atkinson and Donev (1992), Fedorov (1972), Pazman (1986),

Pukelsheim (1994) and Silvey (1980)), and it is difficult to add anything new in

this area of experimental design theory. Theorem 1 which follows, is a gener-

alized version of the Kiefer-Wolfowitz equivalence theorem, (see Kiefer (1959))

and stated here for the reader’s convenience. It also serves as an opportunity

to introduce the notation, which is sometimes different from that used in other

articles of this volume.

Let

where 8 E R” are unknown parameters, fT( z) = (fi(z) . . . , fm(z)) are given

functions, supporting points x; are chosen from some set X , and the Eij are

uncorrelated random errors with zero means and variances equal to one. We

do not make distinctions in notation for random variables and their realizations

when it is not confusing.

For the best linear unbiased estimator of unknown parameters the accumu-

lated “precision” is described by the information matrix:

which is completely defined by the design [ = (.;,pi}?. In the context of the

vi

standard design theory 4

where ( (dz) is a probability measure with the supporting set belonging to X :

suppt c X, and

is the information matrix of an observation made at point x.

Regression model (2.1) and the subsequent comments do satisfy assumptions

1 and 2 from the previous section. To be consistent with assumption 3 let us

introduce a function 9 ( M ) , which is called the “criterion of optimality” in ex-

perimental design literature. A design

is called (@-) optimal.

Minimization must be over the set of all possible probability measures 2 with

supporting sets belonging to X . Now let us assume that:

(a) X is compact;

(b) f(z) are continuous functions in X , f E R”;

(c) Q ( M ) is a convex function and Q ( M ) 5 @(M + A), M 2 0, A 2 0,

Le. matrices M and A are nonnegative definite.

(d) there exists a real number q such that

(e) for any < E Z(Q) and f Z(q):

DESIGN OF SPATIAL EXPERIMENTS vii

where ~(a,<,f) = CY). Here and in what follows we use Q(<) for Q[M([)], Q" for @(<*) and min,,miq, J,

and so on, instead of minzrx, minteE7 Jx, respectively, if it does not lead to am-

biguity.

Theorem 1. . I f (a)-(e) hold, then

1. For any optimal design there exists a design with the same information

matrix which contains no more than n = m(m + 1)/2 supporting points.

2. A necessary and sufficient condition for a design <* to be optimal is fulfill-

ment of the inequality:

miny(x,f*) 2 0. 1:

3. The set of optimal designs is convex.

4. $(x, e") achieves zero almost everywhere in suppt',

where suppt, stands for supporting set of the design (measure) <.

Here and

Functions +(z, <) for the most popular criteria of optimality may be found, for

instance, in Atkinson and Fedorov (1984). Theorem 1 provides a starting point for

analytical exercises with various relatively simple regression problems and makes

possible the development of a number of simple numerical procedures for the

optimal design construction in more complicated and more realistic situations.

Most of these procedures are based on the following iterative scheme:

0 (a) There is a design & E Z(q). Find

x+ = argmin+(z,<,), x- = argmax$(z,Js), XEX XEX,

where X , = suppJs.

... VI11

0 (b) Choose 0 < ps < 1 and construct

[s+l = (1 - P S ) [ , + PS[(XS), where [(s,) is a measure atomized at 2,.

The choice of a sequence ,Os defines a variety of the algorithms; specific examples

are given by Atkinson and Donev (1992), Cook and Nachtsheim (1989), Fedorov

(1972, 1975) and Silvey (1980). The following sequences are most popular:

Theorem 1 together with iterative procedure (2.5), (2.6) provides quite power-

ful tools for constructing optimal design. The existing software products, see, for

instance, Mitchell (1974), Nguen and Miller (1992), Nachtsheim (1987), SAS/QC

Software (1995), Wheeler (1994), confirm this statement. Unfortunately, there

are a few hurdles, which do not allow the direct use of the results reported above.

The first one is that optimal designs defined by (2.4) may have unequal weights.

What does this mean in the context of observing stations allocation? If we have

N available stations or sensors, then r;* = [p:N] stations must be allocated at

s:, where b:lv] is some reasonable integer approximation of pzN. It is obvious

that in many cases (but not always) two or more stations sited in the immediate

vicinity of each other will not give essentially more information than a single

station. There are some arguments in favor of this statement, which can be

expressed economically in colloquial statistical terminology: observations from

these stations are strongly correlated. However, frequently weights pz may be

.

L

DESIGN OF SPATIAL EXPERIMENTS ix

considered as the desirable precision of measurements taken at the i-th station.

The corresponding precision can be achieved through the proper technical steps

or through controlling the longevity of the observational process.

Probably, Gribik et a1 (1976), were the first to use the optimal experimen-

tal design methods for environmental monitoring. They analyzed the problem

of allocating measuring resources to aid in accurately estimating ground level

pollution concentrations throughout a region X . The regression model was the

linearized version of the diffusion model for four pollution sources and unknown

background source. Since the diffusion model used in the study was a large scale

model, measurements separated by distances smaller than a threshold value dis-

tance appeared to be correlated in the corresponding parameter estimation prob-

lem. At the same time the design method was a particular case of the method

discussed in this section, where the independence of observational errors is essen-

tial. To avoid a contradiction the authors imposed the additional constraint: the

distance between any two observing sites must be greater than the characteristic

distance:

(Xi - "j)T("j - Zj) > d2.

Imposing constraints of that type is one of the simplest way to handle possible

correlation between the observed values at neighboring stations. Obviously the

approach does not work for long-range correlation, when the widely separated

observations are correlated.

It was assumed that the ground level pollution is of the prime interest. The

authors proposed to use the weighted average variance of the best linear unbiased

estimator of the ground level pollution:

as the criterion of optimality. The weight function W(Z) was selected proportional

to the population density in the considered region.

A rather detailed discussion of applicability of the standard design technique

x

for spatial experiments may be found in Fedorov et al( 1988).

3. Optimal designs with bounded density

Gribik et a1 (1976), used a very simple and transparent idea to avoid clustering

of sensors at particular points. This idea can be exploited in a more general and

formal setting. Let the number of sensors N be sufficiently large and the density

of stations per square unit be introduced into consideration:

Introduction of (3.1) is very reasonable when the sensor allocation is considered

in technological experiments. In the network allocation problem it is probably

less realistic. Nevertheless, the results considered in this section help to explain

why some intuitive approaches, similar to what was done by Gribik et a1 (1976),

do work well in most cases.

If X is not uniform (as might be appropriate say, with different topography

for different parts of X ) , then it is natural to assume that the sensor density has

to be constrained:

With obvious redefining of the design measure t ( d z ) and the upper bound @2(dz)

the latter may be reduced to a simpler statement:

Thus, the following optimization problem must be considered, (we skip the evi-

dent left hand side constraint):

(* = arg min 8 [ M ( [ ) ] , E

L

DESIGN OF SPATIAL EXPERIMENTS xi

This optimization problem was discussed by Wynn (1982), and Fedorov (1989).

To avoid unnecessary technical complications let us assume additionally to (a)-

(e) from section 2 that

(f) <P(dz) is atomless, Le.

The following theorem summarizes the most important properties of designs

with bounded density.

Theorem 2. . Let ZO be a set of design

((dx) > 0, and ((dx) = 0 otherwise, and let assumptions (a) - (f) hold. Then:

such that t ( d z ) = @(dx), when

- 0 There exists an optimal designs t* E LO.

0 A necessary and sufficient condition for this design to be optimal is that

$(x, [*) separates the two sets X * = supp[* and its complement.

In the above formulation "separate" means that there is a constant C such

that $ ( x , t * ) 5 C on X * and $(z,[+) > C on its complement.

Theorem 1 tells us that supporting sets of optimal designs must coincide with

the points where $(x,[*) achieves its minimum. Therefore, in most cases the

supporting set for the standard optimal design consists of a finite number of

supporting points.

Theorem 2 forces suppc to occupy the subsets of X . How is [*(dx) to be

realized by a practitioner? One of the possibilities is to replace t*(AX) for

relatively small areas AX by N * ( A X ) = [ [* (AX)N] . When N * ( A X ) is defined

then the corresponding number of sensors have to be allocated in A X . For

instance, they can be sited at the nodes of some uniform grid. Generally, that

allocation has to guarantee a reasonable approximation of the integral

xii

by the sum

The properties described by Theorem 2 allow us to formulate a simple numer-

ical algorithm to construct optimal designs (see Fedorov (1989)). Let a(&) =

$(z)dz and

lim a, = 0, lim a,! = 00 and lim < CQ. S+oO S+CO S+oO

s’=l s’=l

(a) There is a design [, E Eo. Let XI, = suppf, and X,, = X \ XIS. Two sets

D, C XI , and E, c X2, with equal measure,

and, correspondingly, including the points

(b) The design ts+l with the supporting set

is constructed.

Usually $(z) is assumed to be constant. All other cases may be converted to

this one with the proper coordinate transformation. In the computerized version

of the algorithm integrals in (a) are replaced with sums over some grid elements.

If these elements and subsequently Q, are fixed and elements of both D, and E,

coincide with the grid elements, then (a), (b) becomes an exchange type algorithm

(see, for instance, Mitchell (1974)) with the simple constraint: every grid element

cannot contain more than one supporting point and the weights of all supporting

points are the same, Le. N-l. In practice it is sometimes convenient to consider

... DESIGN OF SPATIAL EXPERIMENTS Xl l l

grids of varying density, which has to be proportional to #(z). While it can be

shown that the exchange algorithm (a), (b) converges to an optimal design for

properly diminishing a, , it is not generally true for finite a, and, in particular,

when as N - l . The accuracy of the limit designs is defined by the accuracy of

the approximation (see assumption (e) from Section 2):

r

When these approximations are reliable enough then we can hope that the

limit designs do not deviate too much from the optimal ones. The term “limit

design” must be used with some reservation when a, E N-’: instead of conver-

gence some minor oscillations of @ [M(t , )] may be observed. Practical aspects of

the iterative procedure (a),(b) were discussed by Fedorov and Muller( 198913) in

the air pollution network design setting.

4. Correlated observational errors

Let us assume now that the random errors in model (2.1) are correlated and

that the covariance structure is known, i.e. either the covariance matrix V or the

covariance function V ( z, z’) is given. There is no need to use the second subscript

indicating the repeated observations and we consider

where i = 1 , . . . , N , E(&;) 0 and

For the obvious reason, in this section we will use the simplified notation:

xiv

In the case of correlated observations the best linear unbiased estimator is defined

as ( see, for instance, Rao (1973)):

where

where

Unlike (2.2), the information matrix (4.4) is not a sum of information matrices

of single observations. Therefore we cannot use directly the results of the convex

design theory, which is essentially based on the additivity of information matrices.

DESIGN OF SPATIAL EXPERIMENTS

Actually, we have to consider the optimization problem

[G = arg min @ [M(e)J , (4.7) €N

which does not have too much in common with (5) besides notation. For instance,

the convexity of @ is not very helpful anymore.

In most studies authors try to imitate the iterative methods of optimal design

construction considered in two previous sections. For instance, computations

become similar to the standard (uncorrelated) case, if the following recursion

formula is used (Brimkulov et a1 (1986)):

We can easily derive, for instance, that

Subsequently, for the D-criterion the point

must be added to the design &,T. That is an imitation of step (a) from the iterative

procedures considered in the two previous sections.

There exists a simple intuitive explanation why iterative procedures based on

(4.10) provide "good" supporting points in the sense of the D-criterion. First,

let us recollect that in the no-correlation case accordingly to stage (a) of the

iterative procedure from Section 2 the additional observation( s) must be allocated

xvi

at point(s), where the ratio

variance of prediction with the estimated 8 variance of prediction with the given 8

- - a2(x> + fT(z>N-- l (5) f (4 a2(4

is maximal. This follows, for instance, from (2.5) when

(4.11)

in the more general case (see Fedorov (1972)) for details. In the case of correlated

observations we are looking for a maximum of the same ratio

variance of prediction with the estimated 8 variance of prediction with the given 8

When z + xi E s u p p f ~ , then

for f(z) and V ( x , JN) continuous in the vicinity of 2. In other words the iterative

procedure defined by (4.10) does not admit coinciding supporting points. The

result follows from the definitions of +(z,[~) and S2(z,[~), and the fact that

where S;j is the Kronecker symbol. Formula (4.10) can be easily rewritten for

the deleting procedure. LFrom (4.9) it follows that in the case of the D-criterion

(4.12)

DESIGN OF SPATIAL EXPERIMENTS xvii

candidates for deleting are defined by the equation:

We are not aware of any results on the properties of the iterative procedures

based on (4.10) and (4.14) for the D-criterion or similar procedures for other cri-

teria. There are empirical confirmations that the exchange-type algorithms lead

to a significant improvement of the starting design. For instance? Rabinowitz and

Steinberg (1990) applied that type of algorithm to the problem of selecting sites

for a seismographic network. They have shown that the computed designs are rel-

atively efficient and are better than the standard D-optimal designs constructed

for models with uncorrelated observations. It is reasonable to note that compu-

tationally (4.9) and (4.14) are much more demanding than their counterparts in

the standard design theory. There exist a number of studies where the optimiza-

tion problem (4.1) is considered for some special and relatively simple covariance

functions? for instance, generated by autoregressive models. Various details and

further references may be found in Bickel and Herzberg (1979), Bishoff (1992),

Kunert (1988), Martin (1986), Miller and PBzman (1995) and Niither (1985).

In conclusion of this section let us emphasize again the significant difference

between the case with correlated observations and the standard case. For un-

correlated observations the additiveness of the normalized information matrix

(a(x) = 1): IV

A!!([) = N-l f ( x ; ) f T ( x ; ) = N-lM(<N) i=l

leads to many simple and elegant theoretical results initiated by Kiefer’s pioneer-

ing findings. Very frequently normalized information matrices may be treated as

a limit, i.e.:

(4.15)

In many cases for correlated observations the corresponding limit does not exist

and the matrix M ( [ ) cannot be introduced. One of the most successful attempts

xviii

to replace (4.15) was due to by Sacks and Ylvisaker (1966, 1968); see more in

Section 7.

5. Random coefficients regression models: Trend estima-

t ion

In what follows we intend to consider some simple models for the random com-

ponent in (2,l). It is convenient to partition “intrinsic” or “proce~s’~, and “ob-

servational” sources of randomness:

Values u;j describes deviations of the observed response from q(x;, 0) due to

some causes which are independent of an observer. For instance, an average wind

velocity may be disturbed by various local micro-eddies. The term e;j describes

L‘observationa17’ errors. Sometimes these errors are defined by the selected obser-

vational technique and, at least partly, they are controlled by an observer. The

proposed partitioning is very conditional, and the reader may use a different one,

which is more compatible with the corresponding experimental situation.

Let us assume that

or

where

.

DESIGN OF SPATIAL EXPERIMENTS

Vector 0 2 is random with

E(&) = 0, E(620:) = Var(02) = A,

vector e describes the observational errors, which are random and

E ( € ) = 0 E(&) = 021.

We assume that 82 and E are uncorrelated. In terms of (5.1) we have

E ( u ) = 0, E ( U U T ) = F , T ( t N ) A F 2 ( t N ) .

xix

(5.4)

If e = u + E , then

Thus we are going to consider a very special case of (4.1) with V(&v) defined by

(5 .5) . It may be illuminating to associate index “j” with time (hour, day, . . .) and “i” with location (z; is a vector of coordinates of a particular site).

Model (5 .5 ) gives an opportunity to introduce criteria of optimality which

provide a very reasonable description of various experimental situations. Those

criteria may be divided in two main groups. The first group is related to the

“average over time” behavior of the observed response. The corresponding criteria

depend upon the precision of estimators of 01. This means that we consider some

functions of Var(&l), where 81 is an estimator of 61.

The second group deals with “instant” responses and the corresponding cri-

teria are based on V a ~ ( e ) , eT = (e:, 6;).

Let us start with the first group, i.e. with estimating the subvector 61. The best linear unbiased estimator is (compare with (4.3):

xx

The dispersion matrix of 4 is

where

An optimal design (or optimal observational network) is defined as

(5.7)

which differs from (4.7) only by the more detailed information about V ( ~ N ) .

It may be expedient to note that unlike the situation described in comments

accompanying (4.13) the covariance is not anymore a continuous function at the

diagonal:

x+x/ lim E(e(z)e(z ' ) ) # o2 + f . ( ~ ) A f i ( z ' ) . (5.10)

Therefore (5.9) may admit designs with repeated observations, i.e. it could

be that x E supp&v and M ( & + 2) is better than M(JN) .

Using the identity

(5.11)

(5.12)

DESIGN OF SPATIAL EXPERIMENTS xxi

LFrorn the Frobenius formula it follows that

The matrix D ( ( N ) can be also considered as the dispersion matrix of the best

linear unbiased estimator of parameter OT = (e:, O r ) for the regression model.

where F T ( h ) = ( F : ( I ~ N ) , F . ( ( N ) ) , E ( € ) = O,E(ceT) = o21, with the prior

information about parameters 02 described by a prior distribution P(82) such

that

and

See, for instance, Fedorov (1972)? Pilz (1991), and Seber (1977). Thus, the

optimization problem (4.1) may be embedded in the framework of convex design

theory. For instance, for the D-criterion, when (Ml1(&v)I-l = IDl1(&)( must be

minimized, one can use any algorithm developed for the construction of “exact”

or “discrete” optimal designs; see, for instance, Cook and Nachtsheim (1980),

Fedorov (1972), Ermakov (1983), and Pukelsheim (1993) when only the subvector

81 has to be estimated. More generally we can now describe experimental design

as the following optimization problem

(5.13)

where

N is now the total number of possible observations, and

xxii

Let us note that occasionally the total number of observations and the number

of supporting points may coincide (like in (5.2). Then N stands for both. The

results of Sections 2 and 3 may routinely be applied to (5.13) when 8 is properly

defined and o2 and A are known.

Subset D-optimality. According to (4.10) we have to minimize some function

of the matrix a;:, when the parameters 01, are of prime interest. In terms of

(5.13) it means that the objective function 8 must depend upon elements of the

matrix All([), which may be defined as follows:

where

iwOz2 = a2 N - ~ A - ~ .

One of the possibilities is to select

(5.14)

LFrorn Theorem 1 it immediately follows that a necessary and sufficient con-

dition for [* to be optimal is that (compare with Fedorov, 1972):

(5.15)

Notice that

DESIGN OF SPATIAL EXPERIMENTS xxiii

is the normalized variance of JTf(s), where 8 is the best linear unbiased estimator

of 8 from model (5.2), and

may be considered as a normalized variance of the best linear estimator for

the regression model with the same observational errors but with the response

0; f2(x). To get non-normalized values we have to multiply the normalized values

by a2N- l . Simple, but rather long matrix calculations show that

(5.16)

(5.17)

This collection of formulae looks much more complicated than the similar terms

in (5.15). However, that presentation has one remarkable feature: it does not

depend upon functions f2(z) and matrix A explicitly. All elements in (5.16) are

xxiv

completely defined by the covariance function

o - ~ N R ( z , 5') = E (f2(~)6:62f, T I (Z )) = V(Z,Z'), # 5'-

(5.18)

Thus, when the covariance function is known directly, i.e. we do not use (5.5)

to get it, one can use (5.16) and (5.18) to construct optimal design. Moreover,

the cases, in which E = dim62 --+ co may be considered. The first attempt in this

direction was done by Miiller-Gronbach (1993).

r

6. Random coefficient regression models: Prediction

The presentation of the design problem for model (5.2)-(5.4) in the form (5.13)

allows us to develop a rather simple technique for experimental design when the

objective is the prediction of observed values. For the sake of simplicity, let

~ ( z , O ) F 0 in (5.2) and

~ j ( ~ i ) = uij = 6 r f ( ~ i ) .

Then the corresponding optimal designs are defined as

[* = argminQ ( M ( [ ) + Mo) , 5

where

M ( [ ) = / f(z)fT(z)[(dz) and Mo = a2N-lA- l .

It is expedient to note that

DESIGN OF SPATIAL EXPERIMENTS XXV

where the expectation operator E takes into account randomness of both obser-

vational errors and regression parameters. Minimization is taken with respect

to all linear estimators, see Gladitz and Pilz (1982), Fedorov and Miiller (1989),

Pilz (1991) . The best linear estimator is

Similar to arguments in Section 5 we can apply the equivalence theorem to

(6.2) to find, for instance, necessary and sufficient conditions for a design [* to be

optimal. Leaving to the reader the possibility to formulate them for the general

case we focus only on three simple and very popular criteria.

Minimax and D-criterion. For the D-criterion, when

one can easily derive from Theorem 1 that a necessary and sufficient condition

for to be optimal is that

for all x E X .

This inequality appears, especially when the dimension of f is large, more at-

tractive and meaningful in notation described in comments accompanying (5.11):

The variance of the best linear unbiased predictor for u(x) equals

xxvi

= c2 + Cr2N-l [R(z,z) - RT(z7E) (w-1 + R([))-l R(z7E)] ,

where

G(X) = OTf(z) = Rt(2,J) (w-1 + R([))-lP-

= VT(", [) (a2W-1N4 + v(())-I F,

and the components of the vector P are averages of observations at the corre-

sponding points. For the continuously changing weights pi the i-th component of

Y may be considered as the observation made with a precision a/p;N = ai2. If

one introduce the covariance function

-

then (compare with (4.6) or coming later (7.1)) predictor &(z) coincides with

the best linear unbiased predictor for y(z) = u(z) + E(X) everywhere except

x E suppt. At the design points xi the realization(s) of y(z) are measured

directly and are not needed to be predicted, i.e. one may select y(z) = ij(x) and

Var (y(x) - i j(z)I() = 0. Obviously

(6.10)

otherwise.

Using Theorem 1 together with (6.5) and (6.10) we can formulate the analogue

of the Kiefer-Wolfowitz equivalence theorem:

Theorem 3. The following two design problems are equivalent:

There is one significant difference between this result and the original equivalence

DESIGN OF SPATIAL EXPERIMENTS xxvii

theorem: an optimal design generally depends upon the number of observations

N to be used.

Theorem 3 and formula (6.5) give another insight into numerical procedures

from Sections 2 and 3: at every stage one has to relocate the design measure from

the point(s) where y(z) may be predicted easily (small Var(y ( z ) - $(z)[t,)) to

the point(s), where the prediction is poor ( large Var(y(z) - $(x)[&)) . Two linear criteria. Two objective functions which are very popular in spatial

statistics are the weighted average variance of prediction:

and the variance of the weighted average of prediction:

where 2 is the “prediction” set or the area of interest. Using (6.6) one can find

that minimization of Q1(t) and Q2(() is equivalent to minimization of

where in the first case

and in the second one

A = J, w2(z)f(z)fT(z)dz,

A = aaT, a = w(z)fT(x)dz.

LFrom part 2 of Theorem 1 it is easy to conclude that

Theorem 4. . The design e* is linear optimal if and only if

(6.11)

(6.12)

Similar to the D-criterion we can show that for the average variance of pre-

diction

4(z7 5) = 41(3>[) = 1 c0v2(z, Z ' l < ) W 2 ( z 1 ) d Z r , (6.13) z

while for the variance of the weighted average

where

C0v(z7d1t) = R ( z , d ) - RT(.,t) (w-1+ R(J))-I R ( d 7 t ) .

The counterparts of Theorem 3 and 4 may be formulated for optimal designs

with bounded density. To do this function $(z,[*) in Theorem 2 should be

replaced either with Var (y(z) - $(z)It), or with ~I(z, t) , or with +2(z7 5). Remarks on applicability of the results. Let us note that the introduction of

model (5.2) -(5.4) to generate correlated observations allows us to use the convex

design theory for regression problems with correlated observations. Moreover, all

results may be presented in a form which does not demand any direct knowledge

of the functions f(z). We can formulate results for a particular criterion using

only information about the covariance function.

In this and in the previous section we have discussed only the properties of

optimal designs. We hope, that having the sensitivity function $(x, [) represented

for various criteria in terms of the normalized covariance function Cov(z, z ' I ~ ) , the reader can easily construct numerical procedures similar to those discussed

in sections 2 and 3.

7. Comparison with the methods based on the variance - covariance structure of observed random fields

Sacks-Y"visarEer approach. Let us suppose that in model (4.1) there is no trend,

DESIGN OF SPATIAL EXPERIMENTS xxix

i.e. v ( x 7 8 ) E 0, and the covariance function V(x ,x ’ ) is defined and known for

all x,x’ E X . The objective of an experiment is to predict y(x) at a given set of

points 2, which can be either discrete or continuous.,

The best linear unbiased predictor for y(x) may be presented as follows (com-

pare with (4.5 and (4.6)):

We again use notation JN = (x l , . . . ZN) to emphasize that there is only one

observation at every point 2;. Criteria Q 1 ( ( ~ ) and Q2(&) introduced in the

previous section have been most intensively analyzed in the studies related to the

design problem with correlated errors. Usually it has been assumed that 2 = X .

A very good summary of the main results for the criterion Q1((~) may be

found in Micchelli and Wahba (1981). The criterion Q 2 ( & ) was analyzed by

Sacks and Ylvisaker (1970) and Ylvisaker (1987). Further references and com-

ments may be found in Cambanis and Benhenni (1992), Cambanis (1985), Cam-

banis and Su (1993).

Noting (see (7.l)that

where

we can consider minimization of either Q1(&) or & 2 ( [ ~ ) as a problem of finding

an optimal basis for a quadrature formula in approximation theory, with a rather

specific objective function; see Karlin and Studden (1966), Sacks and Ylvisaker

(1970), Stroud (1975).

When N ---f 00, both &I(&) and Q 2 ( [ ~ ) converge to zero for “smooth”

xxx

covariance functions V(z,x') and for any atomless sequence JN. As in Section

3 we may introduce the limit design measure that defines JN. How a sequence

(N may be generated with a particular [ (dz ) is discussed in details by Cambanis

(1985) for X C R1. For instance, the so called regular median sequence or design

.& is defined as

[(dx) = 2N , 2 = l , 2 , . . . , n ,X = [a, b]. (7.3) . X N ; = arg

When X is hypercube and V(z ,x ' ) is separable with respect to all components

of 2, then design [ N may be defined as a direct product of univariate designs (see

Ylvisaker (1975) for details). Thus the design problem is reduced to the search

of the limiting measures providing the best convergence rate for the selected op-

timality criterion. The rather elaborate technique, a close sibling of the classical

approximation theory, leads to a very special minimization problem. Introducing

the design density t ( dz ) = h(s)dx we may state this problem as follows

h* = arg min Q [B(h)] , h (7.4)

where

m is the number of estimated integrals (for instance, integrals of Q2((~)-type

with various weight functions), functions ~,(z) and integer k are defined by a

covariance function and by an optimality criterion.

Similar problems were considered in studies concerned with simultaneous cal-

culation of m integrals by Monte-Carlo method; see Mikhailov and Zhigljavsky

(1989), Zhigljavsky (1988), for details and further references. Actually, (7.4) be-

ing an optimization problem in a space of probability measures has many features

in common with the standard design problem. Some interesting results including

the analogue of the iterative numerical procedure from Section 2 are summarized

DESIGN OF SPATIAL EXPERIMENTS xxxi

and discussed by Zhigljavsky (1988).

. Analytical solutions of (7.4) for the one-dimension case were proposed in the

pioneering papers by Sacks and Ylvisaker (1966, 1968, 1970). Various general-

izations may be found in Hajek and Kimeldorf (1976) and Wahba (1971, 1974).

Following Cambanis (1985), the essence of those findings can be formulated as

follows:

If there exist exactly k quadratic mean derivatives of the random process y(z),

then under certain regularity conditions (see details in the cited publications)

where

and superscripts indicate the order of partial derivatives. For any design with

density separated from zero the integral Q(&,T) diminishes as O(N-2k-2) and

h* (z) minimizes

lim N2k+2 Q ~ ( < N ) ( 7 4 N-bcr,

In fact, the immediateobjective of Sacks and Ylvisaker (1966,1968,1970) was min-

imization of some function of the dispersion matrix of estimators of parameters 6

describing a linear trend 8’f(z) in model (4.1). They reduced the corresponding

minimization problem to minimization of objective functions similar to Q2( &v).

For instance, when 8 f R1, then one has to minimize Q ~ ( [ N ) with a weight

function which is a solution of the following integral equation:

f(z) = J , V(z, z’)w(z’)dz’.

Evidently, for stationary covariance functions ak(z) constant. Subsequently

the optimal limiting density h*(x) is completely defined by the weight function

w(z). In other words, only the behavior of V(z,z’) at its diagonal influences the

solution! The Sacks-Ylvisaker approach (at least in its current form) cannot be

xxxii

used in two cases of practical importance. First, it does not work for “infinitely”

smooth covariance functions, when ak(z) 0 for any IC. The covariance function

is a popular example (see Sacks and Ylvisaker (1966). The presence of the “white”

noise (see model (5.1)) in observed variables gives another example, when the

approach does not work. The latter case is of interest for many applications

being a very reasonable model when a random process is observed with some

instrumental error. In conclusion of this subsection let us note that the concept

asymptotically optimal design [$ based on the existence of a continuous limit

density h*(x) and assumption that

lim max(z;N - = 0;

see Sacks and Ylvisaker (1966). The definition (7.3) of design ,& is one-dimension

by its nature and that makes the approach difficult for spatial applications; see

Ylvisaker (1975) for further details.

N-CQ a

Random parameters approach, To understand the advantages and disadvan-

tages of approach proposed in Sections 5 and 6 relative to to the Sacks-Ylvisaker

approach let us introduce the following model:

where fa(z) are eigenfunctions of the covariance kernel

6, are random with zero means and diagonal covariance matrix, such that Var(6,) =

A,, A1 2 A2 L . . . 2 A, 2 . . . , e(z) is white noise with the variance 1, and o2

is a normalizing constant.

It is well known (Mercer’s theorem; see for instance, Kanwal (1971)) that

DESIGN OF SPATIAL EXPERIMENTS xxxiii

under very mild conditions the series

is uniformly and absolutely convergent and subsequently under very mild as-

sumptions {A,} must diminish not slower then O(l/cr) . For many widely used

stochastic processes or fields the rate is significantly faster; see Micchelli and

Wahba (1989, Theorem 3.

Therefore, for sufficiently large n the kernel Vn(x, 2’) may be a very reasonable

approximation of V ( x , x‘). Allowing 0 +. 0 we may hope that the process yn,@(x)

is “close” to y(x) in the sense of their second moments. Subsequently, we might

expect the closeness of the corresponding optimal designs. This probably holds for

designs 5~ with relatively small N . However, for N + 03 the diminishing Q does

not guarantee closeness of optimal designs with 0 = 0 and 0 > 0. First, formally

the Sacks-Ylvisaker approach does not work for any model with additive “white”

noise, because it causes discontinuity of a covariance kernel at its diagonal:

Secondly, for any cr > 0 and any [ N the rate of convergence for either Q ~ ( [ N ) or

for & ( [ N ) will not be generally better than O(N-’) . This is slower than for any

continuous covariance kernel.

Thus, for large N the Sacks-Ylvisaker approach and results from Sections

5 and 6 may lead to the different asymptotically optimal designs. If one be-

lieves that there is no instrument or any other observation error, then the Sacks-

Ylvisaker approach leads to the better limit designs.

When the contribution of observation errors is significant then approximation

(7.8) becomes very realistic and allows the use of methods from Sections 5 and 6,

which usually produce optimal designs with very moderate numbers of support-

ing points. Usually these designs have about n supporting points.The existence

xxxiv

of well developed numerical procedures and software allows the construction of

optimal designs for any reasonable covariance function V(z7 5’) and various de-

sign regions X, including two and three dimension cases. Let us notice that the

function COV(Z, 2’15) used in Theorem 4 may be presented in the following form:

(7.10)

P = 6;jri7 r; = p;N.

This formula is convenient for some theoretical exercises. For more applied ob-

jectives and for development of numerical algorithms based on the iterative pro-

cedures from Sections 2 and 3, the direct use of eigenfunctions f & ( ~ ) is more

convenient.

Popular IcernekThere are several show-case processes and design regions for

which analytic expressions for the covariance kernel exist, and the corresponding

eigenvalues and eigenfunctions are known:

For the Brownian motion the kernel is

V(Z, z’) = min(z, z’), O 5 x,x‘ 5 1,

and its eigenvalues and eigenfunctions are

A, = (a - 1/2)-’~-’, f,(z) = &sin(a - 1/2)ns, Q = 1,2,. . . .

For the Brownian bridge,

and -2 -2 A, = a R , f,(s) = d i s i n a n z , a = 1,2 ,....

Both kernels are not differentiable on the diagonal (see comments to (7.5)) and

their “jump” functions Q(Z) are easy to calculate. These two kernels or some

DESIGN OF SPATIAL EXPERIMENTS xxxv

simple functionals of them (compare with Wahba (1971)) are convenient candi-

dates for the Sacks-Ylvisaker approach. For the Poisson kernel

1 - p 2

1 - 2pcos27r(J: - xc’) + p 2 ’ V ( X , d ) = 0 2 x,xI 5 1, 0 < p < 1,

and

A* = 1, x 2 , 4 = A2, = pa,

The shape of the Poisson kernel may be controlled by the parameter p. It is

%mooth” at the diagonal and the Sacks-Ylvisaker approach cannot be used.

For most real-world problems it is impossible to represent covariance kernels

in a simple closed form. However, a representation in the form of an infinite series

is standard. For instance, in many experiments related to either diffusion or heat

conduction the covariance kernel may be expressed in the the two dimensional

finite domain case (see Butkovskiy (1982)) as

where X = (0 5 q , x 2 5 l}. Evidently,

fap(s) = 2 sin azrI sin p7rx2, ~ , p = exp [-a2r2(a2 + p 2 ) ] , a, p = I, 2, . . . ,

and a is some constant. Representation (7.11) is very natural and convenient for

the techniques considered in Sections 5 and 6. Note that the physical problems

mentioned above lead to the Gaussian type kernels (compare with (7.7)) when

X becomes infinite with respect to any or both coordinates.

The curious reader will find more covariance kernels in any serious book on

integral equations containing a chapter on definite kernels or describing Green’s

functions (see, for instance, Kanwal (1971) and Butkovskiy (1982)).

Selecting the number of terms in (7.8) sufficiently large assures the closeness

xxxvi

of Vn(z, d) and V ( x , z') may be assured. In many cases it is convenient to assume

that coefficients {e,}? are normally distributed. This assumption does not help

in the present problem and in fact can cause some theoretical difficulties. To

avoid that we suppose that for any Q: distribution P(8,) has a finite support set

[a,, b,] in R1. For instance, we may select some simple symmetric distribution

P ( 0 ) defined on f-a,a] with E(02) = 1 and use P(B,d,) as a distribution for

6,. Selection of distributions with finite supporting sets assures not only close-

ness of an exact kernel and its approximation but proximity of C,"=18afa(z)

and c'& 6,f,(z) if the sequence {f,(s)}? satisfies some routine assumptions

from the approximation theory. The basic idea of using model (7.8) is in deriv-

ing optimal designs for the approximate model 8,f,(z) and verifying the

fact that these designs are optimal or close to optimal for Vn(z,d) or V ( z , d ) .

Furthermore, if the objective function is uniquely defined by a dispersion matrix

of estimated parameters, then the constructed design is optimal for any model

identical to the used one in terms of the first and second moments.

Using (7.8) with o = 0 we may immediately conclude that minimization of

&((AT) is a rather standard problem from the approximate integration theory,

see Davis and Rabinowitz (1985), Stroud (1975).

For instance, it is known (the Gauss-Jacobi Theorem) that for any polynomial

p(x) of degree k 5 2N - 1 the exact equality

.-h N

(7.12)

can be achieved the properly selected weights and supporting points. If { p , ( z ) } f

are orthogonal polynomials with the weight function w(z) > 0,then [AT = {zi)?

are zeros of p j ~ ( z), and

N-1 . -

!IT1 = 4;l(h) = E pi ( . ; ) , (7.13)

DESIGN OF SPATIAL EXPERIMENTS

For example, let us consider the Brownian bridge kernel. Note that

~~

xxxvii

(7.14)

where t = COSTX and Ua-l(t) is the second kind Tschebysheff polynomial. Now,

with z ( t ) = T-’ arccos t , we have for a proper <N = {xi)?

where y,(x) = E:=, OcYfa(x), and it follows from (7.14) that y, ( ~ ( t ) ) / d n is a polynomial of degree not higher than n - 1. If w ( ~ ( t ) ) = d m , then s;V must coincide with zeros of U N ( ~ ) , which are

i N + 1 ’

2; = .(ti) = - i = 1, ..., Nl 2N 2 n;

compare with Muller-Gronbach (1993). Accordingly to (7.13) weights are q ; ( < ~ ) =

T sin2 ~ s i / ( N + l ) , and finally

The solution is extremely simple, but it could be more complicated for other

weight functions, see Davis and Rabinowitz (1986). The value of Q2(&) is of or-

der 0( X2N). Various results about remainders in approximate integration theory

may lead to the better estimates, but the corresponding technique is beyond the

scope of this paper. Further details and related results may be found for instance,

for the one dimension case in Davis and Rabinowitz (1986), Szego (1959) and for

the multi-dimension case in Stroud (1975). Similar exercises may be done for

the criterion & 1 ( e ~ ) with Q = 0. The minimization of Q 1 ( t ~ ) now becomes a

problem from the function approximation theory. When 2 = X and w(z) z 1,

xxxviii

then it follows from (7.1) that

n r \ 2

where vT(x) = V(Z,[N)V-~(&). It is known (see e.g. Micchelli and Wahba (1981)), that

= 2 A,. a=N+1

(7.16)

This lower bound may be used to evaluate the efficiency of JN and can be achieved

for any singular kernel, i.e. when

To verify the latter conjecture one has to select the design [N coinciding with

all zeros of fX(z ) ; see some additional details in Fedorov and Hack1 (1994). In

cases when eigenfunctions cannot be found analytically the use of the remainder

theory is probably one of the most reliable ways to construct satisfactory designs;

see e.g. Davis and Rabinowitz (1984) or Achieser (1956). The ideas discussed

in this subsection help to generate effective designs with very moderate number

of observations N , obviously much less than we need using the Sacks-Ylvisaker

approach based on the local approximation of y(z). The author is not familiar

with any studies where the connection between the classical approximation theory

and the Sacks-Ylvisaker approach were analyzed systematically for models of type

(7.8) with n + 00. Perhaps Micchelli and Wahba (1981) and Miiller-Gronbach

(1993) considered the closest ideas and models.

Again, we would like to note that in most cases measurement errors may

contribute substantially to the randomness of observations. The rule of thumb in

selection of the number of terms in (7.8) is that the least eigenvalue AN should

be significantly less than 0.

xxxix DESIGN OF SPATIAL EXPERIMENTS

8. Discrete case. Optimality criteria and the lower bounds

When the design region X and the set of interest 2 are discrete and contain

Nx and Nz points correspondingly, then the covariance matrix of the vector

{y(x;) - fj(xi)}p, z E 2, completely describes any objective function based

on the second moments. We use the notation D z ( t ~ ) , when the latter matrix

consists of elements

The discrete versions of &I(&v) and & 2 ( t ~ ) are correspondingly

where LT = (1,. . . , 1). We will introduce any weights as we did in the continuous

case, to keep notations simple. In the discrete case we may introduce a very

special version of D-op t imality

It is assumed that there are no points in common for 2 and s u p p t ~ . Otherwise

the determinant equals zero, because

when x; E s u p p t ~ .

Criterion (8.2) is very popular in the statistical literature related to the opti-

mization of monitoring networks; see, for instance, Guttorp et a1 (1993), Carelton

et a1 (1992), Schumacher and Zidek (1993), Shewry and Wynn (1987). In the

cited papers the authors talk about either entropy or information. After the as-

sumption of multivariate normality of the corresponding distributions is made, all

approaches lead to various modifications of D-optimality; compare with Lindley

xl

(1956).

In addition to (8.1) and (8.2) a number of other criteria were introduced for

application in monitoring network improvement. A good collection of them can

be found in Megreditchan (1979, 1989); see also Fedorov and Hackl (1994).

As soon as the criterion of optimality &(<N) (we use this notation to emphasize

that only the criteria of optimality related to the problem of interpolation or

extrapolation are considered in this section) and the kernel V ( z , z’) are defined,

we have to find

E; = arg min &( h). (8.3) t N

It is interestingly to note that optimization problem (8.3) was considered

in a very different setting by Currin et a1 (1991) and by Morris et a1 (1993);

see also Sacks et a1 (1989) for older references. They considered the Bayesian

approach to design of computer experiments and introduced Q(<N) as a measure

of discrepancy between a computer model and its approximation based on some

prior knowledge expressed through the smoothness of the exact response. The

latter was defined by a covariance function.

When N z is relatively “small” and Nx is not very “large” then exhaustive

search may be a proper numerical procedure for a modern computer. With

increase of NZ and NX one can use the exchange type algorithms discussed in

Shewry and Wynn (1987) and in Fedorov and Hackl (1994), which are similar to

those discussed in sections 2-4.

An alternative approach may be based on the introduction of a model similar

to (7.8). For the sake of simplicity of notations, let 2 c X , and let

where XI 2 A2 2 . . . 2 X N ~ , Kj = V(zi,zj) and q , x j E X. Then one may

consider the following approximate model

N

Y 22 YN = e,f, (8-5)

DESIGN OF SPATIAL EXPERIMENTS Xl i

where all vectors Y, YN and fff have Nx components and

Similar to (7.16)

NX

cu=N+l = [(Y-yN)T(Y-yN)] = &Y, (8.6)

where F = (f1,. . . f N ) . There exists another result that can help to evaluate

the closeness of I'v and Y. Let

N

Then (compare with Rao (1973), Ch. 8g)

VN = argmin /IV - All, A

where rank A = N and symbol llBl/ denotes the Frobenius norm of B defined by

(trB2)'i2 = (Eij B;)li2 . Moreover

Thus, the vector YN is the best (maybe not unique) approximation of Y in

the sense of two criteria (8.6) and (8.7). In fact, it is the best one for any strictly

increasing function of D = E [(Y - ?N)(Y - YN)'] which is invariant under

orthogonal transformations, where $$ = BY, rank B 5 N ; see Seber (1984),

Ch. 5.2. The vector

FN = V ( X , [ N ) V - ' ( [ N ) Y 7

where V T ( X , t ~ ) = ( V ( q , 5 N ) ,... ,V(ZN,[N)) , is one of the above linear es-

xlii

timates. Therefore (8.6) and (8.7) help to find the lower bounds for criteria

depending upon

Model (8.5) helps to understand some features of optimal designs and lead to

some interesting numerical procedures (see next section). Adding the “white”

noise, i.e. introducing the following model

N

where E(&) = 0 and E(&‘) = I , allows us to use all the tools discussed in

Sections 2, 3, 5 to generate optimal designs.

9. Unknown covariance function

All the results discussed in the previous sections have essentially used the fact that

either a covariance function V ( z , z’) or a matrix A is known. That is possible but

unfortunately uncommon in practice. In this section we explore two approaches

to estimate the covariance structure.

Direct estimation of a covariance matrix. Let us start with a discrete design

region X and assume there exist repeated observations at every point of X .

Meteorological and environmental networks provide the most typical examples;

see e.g. Megreditchan (1979, 1989) and Oehlert( 1995a,b).

residuals Y - p as Let us define (compare with the previous section) the dispersion matrix of

Where J N = (21,. -. ,zN), y T ( h ) = (~(zI), . . . , y ( z ~ ) ) , and B is an Nx x N

matrix. For the sake of simplicity we assume that E ( Y ) = 0 and that this fact

DESIGN OF SPATIAL EXPERIMENTS xliii

is a priori known. Minimization is understood in the matrix ordering sense. A

solution of (9.1) is

B* = v(x,tN)v-l(tN)

When sufficiently many observations are accumulated at every point of X the

strong law of large numbers assures us that

(9-2) and subsequently B* and B*, which minimize correspondingly the left and right

hand-sides, are close to each other. Straightforward minimization gives

where both matrices with caps are evident partitions of

k

Q ( X ) = q y . j=1

(9-4)

When there are missing observations, then it is better to use instead of (9.4)

pairwise estimates

where

and

kil

j=1

& ( X ) = ICif'Cy,iT(,l,

IC;! is the number of cases when the response variable was measured at xi

simultaneously.

Thus, (9.2) - (9.4) lead us to a very simple and widely used recipe: replace

unknown parameters by their estimates and use methods developed for cases in

which all parameters are known. Together with this simple recommendation (9.2)

helps to generate other versions of numerical algorithms considered earlier. Let

xliv

us introduce matrix I ( J N ) with the following elements:

Sii, when x; f supp J N ,

0, otherwise. IiZ(<N) =

The left-hand side of (9.2) may be represented now as

and the design problem may be viewed now as

iFrom the numerical point of view (9.5) may be considered as a multi-dimension

version of the best regression selection problem. Stepwise regression and best

subset selection are the popular algorithms and can be easily adopted to solve

(9.5). In fact the same methods may be used when the matrix f i(Sp~,B) is

replaced by its true value; see comment in the conclusion of Section 8. Let

i.e. we want to minimize the variance of prediction at point z1. In this case

and it is a very standard problem of selection of N predictors from Nx - 1

candidates and there exist a numerous number of the statistical packages which

can be used to do that. The author is not familiar with multi-dimension versions

of the corresponding software products, which are needed for more complicated

criteria. The idea to use the least squares technique for selection of the most

informative subset of sensors was probably initiated by Megreditchan (1979).

.

DESIGN OF SPATIAL EXPERIMENTS xlv

The search for an optimal design &,J may be viewed in this setting as a parti-

tioning of X into two sets of given size NX - N and N . The latter must contain

the most information about the whole set X; see, for instance, Shewry and Wynn

(1987), who proposed using

where the subscript “p” indicates that the matrix contains only elements corre-

sponding to the points (sites) with no observations. When (9.7) is replaced by

its empirical version

then the following simple and intuitively attractive exchange-type procedure may

be used to construct &; see Fedorov and Hack1 (1994):

(a) Given ( N ~ = { x i s } ; find

k N

i+ = arg maxmin

Add the point x;+ to the design: [ ( N + I ) ~ = [ N s + xi+. (b) Find

where xz E J ( N + ~ ) ~ and delete the point 2;- from the design, i.e. construct

t N ( s + l ) = <(N+l)s - 3 i - e Retun to (a).

Briefly, the exchange procedure (a), (b) may be spelled out in the following

way: add to the design the worst explained sites and delete from it the best

explained sites. Apparently, the approach may be called “model free”: only

existence of first two moments of observed Y is assumed. That may attract many

practitioners. However in the search for an optimal network we are confined to

sites where the measurements have been previously made. In other words, the

xlvi

selection of the most informative subsets of sites (sensors, observing stations)

may be discussed, but we cannot consider the problem of optimal extension.

Estimation of a parameterized covariance. In many practical cases the design

region X is a continuous set and the covariance function has to be known every-

where at X . The most popular approach is based on the assumption that this

function is homogeneous and isotropic, i.e.

V ( z , d) = V ( r ) , T 2 = (3 - .E')T (z - z')

with the subsequent parsimonious approximation of function V ( r ) ; see e.g. Cressie

(1991), Marshall and Mardia (1985), M a t h (1986), Ying (1995). The approach

is frequently used in geostatistics, where a single realization of a random field is

available, and in particular in the "kriging method" paradigm.

Methods from in Sections 5-8 are essentially based on approximation of the

observed random fields by regression models with random coefficients. When

prior to design of a network there exist some historical observations, then one

may use the technique, which was developed for these models.It is expedient to

note that accurate knowledge of A or A, is useful but it is not as crucial as the

knowledge of a covariance function in the Sacks-Ylvisaker approach. In fact, in

basic optimization problems (5.13) and (6.2) the objective functions depend upon

the sum M(<) + Mo, where Mo is defined by A. For instance, in the case of (6.2)

and therefore the role of A diminishes when either a2 --+ 0 or N + 00. Moreover,

the simple dependence upon A allows to construct numerically optimal designs

for different matrices A to learn about their sensitivity with respect to A.

In the simplest case, when the observational errors are negligible, the following

estimators may be used:

j=1

DESIGN OF SPATIAL EXPERIMENTS xlvii

where 8 f E", X j is the set of points with observations yj(z;) , and

for all j = 1,. . . , IC. It is assumed that functions f(z) are known and

Subsequently,

Actually, it is more convenient to use the matrix An directly than the function

p ( x , 2') in all numerical procedures discussed in Sections 5 , 6.

When the observational errors are comparable with the variations of 6, then

(9.9) must be replaced with more sophisticated estimators, which are computa-

tionally much more demanding and complicate. Details and references may be

found in Spjotvill (1977) and Fedorov et al. (1993).

10. Space and time

In most spatial experiments, after the sites are selected measurements are usually

taken on some regular schedule, for instance, several times a day, or they are

continuously recorded. Generally, the response function may depend upon time.

Random errors can be correlated both in time and space. We consider only

the simplest case, where there is no spatial correlation, following the ideas from

Section 2. The generalization for more general models considered in Sections 3-6

is straightforward.

To adopt (2.1) for the time dependent response we assume that

xlviii

and

When p j j t = 6 j j t , then the information matrix of observations made accord-

ingly to the time schedule ( (d t l z ) may be presented in the following form

For measurements which are correlated in time,

where, for the sake of simplicity, we assume that supp((dt/z) is a discrete set

tl, t 2 ... 7 t77 and R(z ) = p j j l ( z ) i -

When the measure ( ( d t l z ) is fixed for each given z, then all the results from

Section 2 may be used, with obvious replacement the function $(z, t), which in

the standard case has the form

for all criteria satisfying assumptions (a) - (e), by the function

G(2 , t ) = @(e> - trm(z)A(S)*

For instance, for the D-criterion the sensitivity function m - f T ( z ) M - ' ( [ ) f ( z )

must be replaced by m - trm(z)M-'([). More details may be found in Atkinson

and Fedorov (1988), Fedorov and Nachtsheim (1995), and Spruil and Studden

(1979). Formally the time dependent observation may be treated as a vector-

observation case (see, for instance, Fedorov (1972), Ch. 5).

Evidently, introducing the time variable does not change the basic theory,

but makes all techniques, including computing of optimal designs, more time and

effort consuming. However there exist models and optimality criteria for which

DESIGN OF SPATIAL EXPERIMENTS xlix

optimal designs are the same both for the static and for the time dependent

cases. For instance, the latter is true for models with uncorrelated observations

and with separable variables, when

or where

and the selected criterion satisfies assumptions ( a ) - ( e ) from Section 2; see Cook

and Thibodeau (1980), Hoe1 (1965), Huang and Hsu (1993), Schwabe (1994,

1995).

When time is included explicitly in model, then the concept of sensor alloca-

tion can be extended and "mobile" sensors may be introduced. In this case design

consists of trajectories x;(t) f X,O 5 t 5 2'. The topic is beyond the scope of

this survey. A reader can find the results and references in Chang (1979), Fedorov

and Nachtsheim (1995), Titterington (1980) and Zarrop (1979).

Acknowledgement

I am most grateful to my immediate colleagues D. Downing and M. Morris for

their very constructive and effective help in preparing this paper. I thank B.

Wheeler for his numerous and very useful comments and suggestions.

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